My instructor introduced the definition of permutation and symmetric group as follow:
Permutation group: On an arbitrary set $X$, the permutation group is all the bijection maps: $X\to X$.
Symmetric group: On a finite set $X=\{1,2,...,n\}$, the symmetric group is all the bijection maps $X\to X$. Based on this definitions, it appears to me that symmetric group and permutation group are really the same things except the following:
- One is permuting numbers and the other is permuting any arbitrary elements.
- Symmetric group is referring to finite set.
He mentioned that symmetric group is a subgroup of permutation group where symmetric group is only working with finite numbers. He also proofed a theorem that permutation group is isomorphic to symmetric groups, which appears fine to me based on his definition of groups but not ok on the definitions I found elsewhere.
Based on my understanding, a more standard way of thinking permutation group is to consider them as a subgroup of symmetric group where only contains SOME but NOT ALL permutations on a given set. Refer to this question.
This two kind of definitions seem very different to me and I'm not sure how I can connect these two together. Need clarification on whether this is just two different ways of understanding them or one of them is incorrect.
You're correct in that permutation groups don't have to contain all permutations on $X$.
A permutation group on $X$ is a subgroup of the symmetric group on $X$. (however there can be many such permutation groups, but only one symmetric group)